[Mika]

Well, how much is almost all and how much smoothness is enough?

This is the question Physicist has to answer. As a result, he either becomes a hero or a laughingstock.

As I said earlier, in some contexts a failure of analyticity is considered a singularity, and analyticity often fails somewhere with many physical situations (even say like, walking around a room).

But that has never stopped Physicists from doing it anyway - or they never thought about it and then it never bothered them. Sometimes I think it is good not to know too much about Maths. Take a look at the history of the scalar diffraction theory by Kirchoff for example. In short, Kirchoff formulated the theory with a quite grave mathematical error, but the theory gave correct results anyhow. The modification by Sommerfeld (and Rayleigh?) and some other people put it on the sound mathematical basis, but generality was restricted.

Mika

[CP5670]

But that has never stopped Physicists from doing it anyway - or they never thought about it and then it never bothered them.

What is "it?"

I'm just disputing your original statement that everything has perfect regularity and singularities never show up in real life. Whether that bad behavior actually poses a problem or not depends on the context. In many cases, the mathematical framework is set up to allow the types of irregularities you encounter and you don't need to do anything special about them.

Sometimes I think it is good not to know too much about Maths.

I think this Hilbert quote is oddly appropriate here: "Physics is too hard for physicists."